hello, I could give answer to your first question

when a square is inscribed in a circle, the length of the diagnol of the square =the diameter of the circle

2^(1/2)side of square = diagonal of square

area of square = 1/2 * diagonal * diagonal

S1 = 2 (diagonal of S1 =2)

when a circle is inscribed in a square, length of square = diameter of circle

diameter of O2= length of S1 = 2^(-1/2)* diagonal of S1 = 2^(1/2)

S2 = 1/2 * 2 = 1

Sn = 2*[(1/2)^(n-1)]